Question

Which is the interval notation represent the domain?

Answer 1

Answer:

The first choice, $(-\infty, 7) \cup (7, \infty)$.

Step-by-step explanation:

The dashed vertical line is a vertical asymptote. The x-coordinate of all points on this vertical asymptote are equal to 7. In other words, when the x-value of a point on the graph approaches $7$, its y-value approaches infinity (or negative infinity.)

Either way, the graph is not defined for $x = 7$. The point $7$ should thus be excluded from the domain of the graph.

The graph is apparently defined for all other x-values. The domain of the function should thus be all real numbers with the exception of $x = 7$. Here's how to write that in interval notation:

• The set of all real numbers can be expressed as the interval $(-\infty, \infty)$. Note that infinity $\infty$ (or $(- \infty)$ itself isn't a real number. The round brackets indicate that both endpoints are excluded from the from the interval.
• The set of all real numbers less than (not equal to) $7$ is $(-\infty,7 )$ (both ends are excluded.) The set of all real numbers greater than (not equal to) $7$ is $(7, \infty)$.
• The set of all real numbers that is not equal to $7$ is the union of all real numbers less than $7$ and all real numbers greater than $7$. The union operator $\cup$ connects two intervals.

In other words, the domain is $(-\infty, 7) \cup (7, \infty)$.